

A181293


Triangle read by rows: T(n,k) is the number of 2compositions of n having k 0's (0<=k<=n) A 2composition of n is a nonnegative matrix with two rows, such that each column has at least one nonzero entry and whose entries sum up to n.


1



1, 0, 2, 1, 2, 4, 2, 6, 8, 8, 4, 14, 24, 24, 16, 8, 32, 64, 80, 64, 32, 16, 72, 164, 240, 240, 160, 64, 32, 160, 408, 680, 800, 672, 384, 128, 64, 352, 992, 1848, 2480, 2464, 1792, 896, 256, 128, 768, 2368, 4864, 7296, 8288, 7168, 4608, 2048, 512, 256, 1664, 5568
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OFFSET

0,3


COMMENTS

The sum of entries in row n is A003480(n).
Sum_{k=0..n} k*T(n,k) = A181294(n).


LINKS



FORMULA

G.f.: G(t,z) = (1z)^2/(12*z2*t*z+2*t*z^2).
G.f. of column k is 2^k*z^k*(1z)^{k+2}/(12*z)^{k+1} (we have a Riordan array).


EXAMPLE

T(2,0)=1, T(2,1)=2, T(2,2)=4 because the 2compositions of 2, written as (top row/bottom row), are (1/1), (0/2), (2/0), (1,0/0,1), (0,1/1,0), (1,1/0,0), (0,0/1,1).
Triangle starts:
1;
0,2;
1,2,4;
2,6,8,8;
4,14,24,24,16;
...


MAPLE

G := (1z)^2/(12*z2*t*z+2*t*z^2): Gser := simplify(series(G, z = 0, 14)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



